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Alternating series are series whose terms alternate in sign between positive and
negative. There is a powerful convergence test for alternating series.
Many of the series convergence tests that have been introduced so far are stated with
the assumption that all terms in the series are nonnegative. Indeed, this condition is
assumed in the Integral Test, Ratio Test, Root Test, Comparison Test and Limit
Comparison Test. In this section, we study series whose terms are not assumed to be
strictly positive. In particular, we are interested in series whose terms alternate
between positive and negative (aptly named alternating series). It turns
out that there is a powerful test for determining that a series of this form
Let be a sequence of positive numbers. An alternating series is a series of the form
or of the form
As usual, this definition can be modified to include series whose indexing starts
somewhere other than .
The geometric series
is alternating. In general, a geometric series with ratio is alternating, since
Recall the harmonic series
which is perhaps the simplest example of a divergent series whose terms approach
zero as approaches . A similarly important example is the alternating harmonic
The terms of this series, of course, still approach zero, and their absolute values are
monotone decreasing. Because the series is alternating, it turns out that this is
enough to guarantee that it converges. This is formalized in the following
Alternating Series Test Let be a sequence whose terms are eventually positive and
nonincreasing and . Then, the series both converge.
Compared to our convergence tests for series with strictly positive terms, this test is
strikingly simple. Let us examine why it might be true by considering the partial
sums of the alternating harmonic series. The first few partial sums with odd index
are given by
Note that a general odd partial sum is of the form
and the quantity in the parentheses is positive. We conclude that:
The sequence of odd partial sums defined above is
Moreover, the sequence of odd partial sums is bounded below by zero, since
and each quantity in parentheses is positive.
We conclude that the sequence of odd partial must converge to a finite limit by
The Fundamental Theorem of Calculus.The Monotone Convergence
Theorem.The Ratio Test.
Next consider the sequence
of even partial sums.
The sequence of even partial sums defined above is
increasing and bounded, and
therefore converges by the Monotone Convergence Theorem.decreasing and
bounded, and therefore converges by the Monotone Convergence Theorem.
Finally, we use
and the fact that the limits of the sequences and are finite to conclude
It follows that the alternating harmonic series converges.
With slight modification, the argument given above can be used to prove that the
Alternating Series Test holds in general.
Does the series converge?
The terms of the sequence are positive and nonincreasing, so we can apply the
Alternating Series Test. Since
the Alternating Series Test implies that the series converges.
Does the alternating series test apply to the series
The underlying sequence is . This sequence is positive and approaches as . However,
it is not a decreasing sequence; the value of oscillates between and as . We cannot
remove a finite number of terms to make decreasing, therefore we cannot apply the
alternating series test.
Keep in mind that this does not mean we conclude the series diverges; in fact, it does
converge. We are just unable to conclude this based on the alternating series test.